Scattering From an Elliptic Crack by an Integral Equation Method: Normal Loading

2002 ◽  
Vol 69 (6) ◽  
pp. 775-784
Author(s):  
T. K. Saha ◽  
A. Roy
Keyword(s):  
Integral Equation ◽  
Crack Opening Displacement ◽  
Cross Section ◽  
Infinite System ◽  
Crack Opening ◽  
Scattering Cross Section ◽  
Elliptic Crack ◽  
Opening Displacement ◽  
Normal Loading ◽  
Scattering Cross

The scattering of normally incident elastic waves by an embedded elliptic crack in an infinite isotropic elastic medium has been solved using an analytical numerical method. The representation integral expressing the scattered displacement field has been reduced to an integral equation for the unknown crack-opening displacement. This integral equation has been further reduced to an infinite system of Fredholm integral equation of the second kind and the Fourier displacement potentials are expanded in terms of Jacobi’s orthogonal polynomials. Finally, proper use of orthogonality property of Jacobi’s polynomials produces an infinite system of algebraic equations connecting the expansion coefficients to the prescribed dynamic loading. The matrix elements contains singular integrals which are reduced to regular integrals through contour integration. The first term of the first equation of the system yields the low-frequency asymptotic expression for scattering cross section analytically which agrees completely with previous results. In the intermediate and high-frequency scattering regime the system has been truncated properly and solved numerically. Results of quantities of physical interest, such as the dynamic stress intensity factor, crack-opening displacement scattering cross section, and back-scattered displacement amplitude have been given and compared with earlier results.

SCATTERING BY A NARROW UNIDIRECTIONALLY CONDUCTING INFINITE STRIP

1960 ◽  
Vol 38 (12) ◽  
pp. 1623-1631 ◽  
Author(s):  
S. R. Seshadri
Keyword(s):  
Integral Equation ◽  
Electromagnetic Wave ◽  
Cross Section ◽  
Plane Electromagnetic Wave ◽  
Scattering Cross Section ◽  
Wave Number ◽  
Infinite Strip ◽  
Total Scattering ◽  
Scattering Cross ◽  
Total Scattering Cross Section

The scattering of a plane electromagnetic wave of wave number k by a uni-directionally conducting infinite strip of width 2a is investigated, The problem is formulated in terms of an integral equation whose solution is obtained by a well-known procedure in the form of a series in powers of ka. Expressions for the far-zone fields and the first two terms in the series for the total scattering cross section are obtained.

Diffraction of Antiplane Shear Waves by an Edge Crack

1980 ◽  
Vol 47 (2) ◽  
pp. 359-362 ◽  
Author(s):  
S. F. Stone ◽  
M. L. Ghosh ◽  
A. K. Mal
Keyword(s):  
Integral Equation ◽  
Crack Opening Displacement ◽  
Edge Crack ◽  
Incident Angle ◽  
Crack Opening ◽  
Shear Waves ◽  
Antiplane Shear ◽  
Opening Displacement ◽  
Low Frequencies ◽  
High Frequencies

The diffraction of time harmonic antiplane shear waves by an edge crack normal to the free surface of a homogeneous half space is considered. The problem is formulated in terms of a singular integral equation with the unknown crack opening displacement as the density function. A numerical scheme is utilized to solve the integral equation at any given finite frequency. Asymptotic solutions valid at low and high frequencies are obtained. The accuracy of the numerical solution at high frequencies and of the high frequency asymptotic solution at intermediate frequencies are examined. Graphical results are presented for the crack opening displacement and the stress intensity factor as functions of frequency and the incident angle, Expressions for the far-field displacements at high and low frequencies are also presented.

Elastodynamic Analysis of a Periodic Array of Mode III Cracks in Transversely Isotropic Solids

1992 ◽  
Vol 59 (2) ◽  
pp. 366-371 ◽  
Author(s):  
Ch. Zhang
Keyword(s):  
Integral Equation ◽  
Crack Opening Displacement ◽  
Boundary Integral Equation ◽  
Crack Opening ◽  
Transversely Isotropic ◽  
Boundary Integral ◽  
Crack Spacing ◽  
Periodic Array ◽  
Mode Iii ◽  
Opening Displacement

Time-harmonic elastodynamic analysis is presented for a periodic array of collinear mode III cracks in an infinite transversely isotropic solid. The scattering problem by a single antiplane crack is first formulated, and the scattered displacement field is expressed as Fourier integrals containing the crack opening displacement. By using this representation formula and by considering the periodicity conditions in the crack spacing, a boundary integral equation is obtained for the crack opening displacement of a reference crack. The boundary integral equation is solved numerically by expanding the crack opening displacement into a series of Chebyshev polynomials. Numerical results are given to show the effects of the crack spacing, the wave frequency, the angle of incidence, and the anisotropy parameter on the elastodynamic stress intensity factors.

Mass Determination by Direct Measurement of Electron Scattering

1975 ◽  
Vol 33 ◽  
pp. 240-241
Author(s):  
M. K. Lamvik ◽  
A. V. Crewe
Keyword(s):  
Cross Section ◽  
Electron Scattering ◽  
Mass Density ◽  
Variable Mass ◽  
Scattering Cross Section ◽  
Mass Determination ◽  
Number Density ◽  
Cross Sectional ◽  
Thin Slice ◽  
Scattering Cross

If a molecule or atom of material has molecular weight A, the number density of such units is given by n=Nρ/A, where N is Avogadro's number and ρ is the mass density of the material. The amount of scattering from each unit can be written by assigning an imaginary cross-sectional area σ to each unit. If the current I0 is incident on a thin slice of material of thickness z and the current I remains unscattered, then the scattering cross-section σ is defined by I=IOnσz. For a specimen that is not thin, the definition must be applied to each imaginary thin slice and the result I/I0 =exp(-nσz) is obtained by integrating over the whole thickness. It is useful to separate the variable mass-thickness w=ρz from the other factors to yield I/I0 =exp(-sw), where s=Nσ/A is the scattering cross-section per unit mass.

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